For questions involving random variables uniformly distributed on a subset of a measure space. To be used with [probability] or [probability-theory] tag.

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70 views

Interval of non-uniformly distributed set of numbers adjusted that it properly excludes extremes

Let's say I have an interval of numbers from 1 to 9 with the following frequency of distribution: numbers 1, 2 and 3 about 20 occurrences number 6 has 2 occurrences and number 9 has only ...
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1answer
35 views

Probability that at least one event (out of two uniform RV) happens before two other random events

I recently faced a probability problem that is puzzling me. I would like to ask you if you could help me. I have two random variables X1 and X2 i.i.d with uniform distribution U[64,96] and other two ...
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1answer
89 views

How to find the pdf of the minimum of absolute differences of Uniform distributions.

Let $X_1$,$X_2$ and $X_3$ are independent random variables that are uniformly distributed over $(0;b), b>0$. What is the probability density function of z=min($Y_1$,$Y_2)$, where $Y_1=|X_1-X_2|$ ...
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1answer
42 views

Looking for a simple bivarate uniform distribution with non-zero covariance matrix

Obviously there are many forms this can take, I'm looking for on that gives an non-zero (off diagonal elements) covariance matrix. Does anyone know of one?
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1answer
31 views

Uniformly Distributed ingredients

Suppose we need to make a dish that has three ingredients A, B and C. All are distributed uniformly between [0, 2], [0, 2], [0, 1] respectively. To create the dish, we need 1/4 of A, 1/4 of B and 1/8 ...
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1answer
28 views

Finding the joint density of two random variables

Suppose (X,Y) is uniformly distributed over the region { (x, y) : 0 < x < y < 1 }. Find the joint density of (X, Y). I started out by drawing the unit square and filling in the area where 0 ...
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71 views

How to construct a uniform joint distribution

I have a question that is critical to my work, but I am not sure if it is any possible. Assume that you have two uniform random variables X and Y. The product distribution of Z=XY is not a uniform. ...
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1answer
52 views

Finding probability of uniform random variable given a condition with another random variable

Suppose X and Y are independent and uniformly distributed on the unit interval (0,1). Find: $$P[Y>\frac{1}{2}\,|\,Y>1-2X]$$ How I approached it was to find the area where $Y > 1 - 2X$, and ...
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1answer
145 views

Probability of maximum of 2 uniform random variables

The random variables X and Y are independent, each with the uniform distribution on [−1, 1]. Find: $$P[max (X,Y) >0.5]$$ Apparently there is an easy approach without integration, but I am having ...
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0answers
37 views

choose a list of words such that have equal letter frequency

I have a big list meaning full Words. surely letter frequency of this word list is different for each letter. Now my problem is to find a way to randomly select words from this word list to a new ...
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2answers
201 views

Expected value of the function of a uniformly distributed random variable

Let X by a uniformly distributed random variable on the interval [0,1]. Find $E[e^Y]$ I am trying to make use of the formula $$E[g(X)] = \int_{-\infty}^{\infty}g(x)xdx$$ so then $$E[e^X] = \int ...
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480 views

What is the expected time you have to wait until the first bus comes?

 three buses, bus A, B, and C come to a bus stop every hour. The time at which each bus arrives at the stop is distributed as a uniform random variable, i.e., TA,TB,TC ∼ Unif[0,1] hours. The ...
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1answer
69 views

$P\{B^{2}-4AC\geq 0\}$ where $A,B,C \sim U(0,1)$?

The actual problem is to find the probability that $Ax^{2}+Bx+C=0$ has real roots. This boils down to whether or not the discriminant $B^{2}-4AC$ is non-negative. Thus, we seek $P\{B^{2}-4AC\geq 0\}$. ...
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0answers
96 views

Summing many non-standard i.i.d. uniform random variables

all! I have looked up a fair bit on this question and learned much about the problem. But haven't been able to get any crisp answers. Sorry, if I'm missing something obvious. I know one can use the ...
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1answer
55 views

What is the distribution of $X+Y$ where $X \sim U(0,\frac{L}{2})$ and $Y \sim U(\frac{L}{2},L)$?

I started along these lines: Let $Z = X + Y$ where $\frac{L}{2}< z < \frac{3L}{2}$, then, $$f_{X+Y}(z)=f_{Z}(z) = \int f_{X}(x)f_{Y}(z-x)dx$$ However, I am not sure how to fill in the bounds ...
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4answers
285 views

Find the probability of $a>b+c$, where $a$, $b$, $c$ are $U(0,1)$

What is the probability that $a > b + c$? $a, b, c$ are picked independently and uniformly at random from bounded interval [0,1] of $\mathbb{R}$.
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788 views

Probability Integral Transformation

I just attended an introductory course on Statistics and we came across the following: I know what random variables, the uniform distribution, etc. are but the notation from the proposition ...
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0answers
80 views

Uniform distribution

You arrive at a bus stop at 10'0 clock, knowing that the bus will arrive at some time uniformly distributed between 10 and 10:30. what is the probability that you wait longer than 10 minutes? if at ...
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1answer
23 views

What is the distribution of the impact point of a Random Ray

In the $\displaystyle (O,x,y)$ plane, a random ray emerges from a light source at the point $\displaystyle (-1,0)$, towards the $\displaystyle (O,y)$ axis. The angle with the $\displaystyle (O,x)$ ...
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1answer
166 views

Linear Combination of Min and Max of Uniform Random Variables

Let $p$ and $q$ be uniformly distributed on $[0,1]$. Define $x=\min\{p,q\}$, $y=1-\max\{p,q\}$ and $z=1-x-y$. What are the distribution functions of $x$,$y$ and $z$? I've got $F_X(x) = 1 - (1-x)^2$ ...
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1answer
32 views

Finding variance .

Suppose that $f : [0, 1] → [0, 1]$ and we wish to estimate $$I = \int_{0}^{1} f(x) dx$$ Using the hit-and-miss method, we obtain the estimate $$\hat I_{HM}=\frac{1}{n}\sum_{i=1}^{n}X_i$$ where ...
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1answer
156 views

Plot the cdf and simulate a random variable (rv) with this cdf using the inversion method.

Consider the continuous random variable with pdf given by: $$f(x) = 2(x − 1)^2;\quad 1 < x ≤ 2$$ $$f(x) = 0;\quad \text{otherwise}$$ Plot the cdf for this random variable. Show how to simulate ...
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2answers
56 views

$X$ is half normal and $S ∼ U{(−1, +1)}$. How $Z = SX ∼ N(0, 1)$?

If we chop a standard normal distribution in half and use only the positive side (scaled up by a factor of $2$ to maintain a proper density), then we get the so-called ‘half normal’ density: ...
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2answers
385 views

Uniform Distribution : pdf & inverse cdf

$X\sim U(1,3)$. Verify that X has cdf $F_X(x) = 2(x − 1)$ for $x \epsilon(1, 3)$ and thus that $F^{−1}_X (y) = 2y +1$ for $y \epsilon (0, 1)$. My attempt for ...
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1answer
190 views

Question regarding finding Joint distribution of two random variables

I have a question regarding finding the following joint distribution. Let $p \sim U[0,1]$, standard uniform distribution. The random variable $X$ is defined as $X = 2$ with probability $p$ and $X ...
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1answer
2k views

Probability density of Continuous uniform distribution over the unit circle

If we want to chose a point $(x,y)$ uniformly at random from a unit circle in a plane, why is the joint probability density of the random variable $f(x,y) = \frac{1}{\pi}$ for $x^2+y^2\leq1$? The ...
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1answer
183 views

Convolution of Discrete Uniform ,$DU$, Distribution.

If $X\sim DU(k,a,h),\quad -\infty<a<\infty,h>0=1,2,\ldots$ then the probability function is $$P(X=a+jh)=\frac{1}{k},\quad j=0,1,\ldots,k-1$$ Let $Z\sim DU(r,0,s)$ and $Y\sim DU(s,0,1)$ , ...
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1answer
298 views

Finding a PMF with random variables

Let $X$ be a discrete random variable that is uniformly distributed over the set of integers in the range $[a,b],$ where $a$ and $b$ are integers with $a<0<b$. Find the PMF of the random ...
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1answer
83 views

spin arrow of random variables

Spin an arrow attached to the center of a circular board, let theta be the final angle of the arrow, theta<= 2pi. The probability that theta falls in a subinterval (0, 2pi] is proportional to ...
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1answer
41 views

On uniform number generation with vectors

Let $\vec{a}$ be a random unitary vector. If $\vec{\lambda}$ is a uniformly distributed vector on $\mathbb{S}_2$ (the unitary sphere?), could we say that the result $|\vec{a}.\vec{\lambda}|$ is ...
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0answers
101 views

Expected number on the convex hull [closed]

My question is not very difficult, but I have problem with that : Prove that if S is the set of n points sampled from a uniform distribution in a unit square, then the expected number of points on ...
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1answer
62 views

Equivalence of uniform distribution

Behind a rectangle grid evenly (i.e. uniform distribution) scattered dots. Could it be considered identical (will have the same uniform distribution) to a sequence of independent events with ...
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2answers
48 views

Number of times you have to make a bet on a uniform distribution to expect to achieve a minimal result

Edited for the sake of clarity: If you have a random variable $Q$ distributed uniformly on some interval, say $[a,b]$, what is the function $f$ that describes how many times you have to draw on the ...
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1answer
55 views

How Do I Get This Joint Density Function?

Given $X \sim u(0,1)$, we define $Y=1-X$, then we have that $f_{X}(x)=I_{[0,1]}(x)$ and $F_{X}(x)=xI_{[0,1]}(x) + I_{(1, \infty)}(x)$. I know, if $0\le y \le 1$ $$F_{Y}(y)=P[Y \le y]=P[1-X \le ...
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0answers
44 views

Is $e$ uniformly distributed in all bases?

There has been talk of whether or not $\pi$ is normal, i.e. uniformly distributed in all bases $b$ where $b\ge2$. The general response has been that we expect that it is, and have found no obvious ...
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0answers
190 views

Max variance of uniform distribution?

Suppose I roll a 20-sided die 1000 times and count the number of times a particular value comes up. This gives an array of 20 counts, and the expected value of each is 1000/20 = 50. I'd like to find ...
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1answer
549 views

Range of Uniform Distribution

I'm trying to compute the density for the range $R_n$ for samples of a random variable $X$ that are uniformly distributed on the interval $(a,b)$. We define the range as $$ R_n = X_{(n)} - X_{(1)}, ...
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2answers
188 views

Random Variables from $[0,1]$ - Integration Limits

I was wondering if someone could help me understand the first steps I should take for solving the next problem: Let $U$, $V$ be random numbers chosen independently from the interval $[0, 1]$ with ...
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1answer
117 views

Sum of Two (General) Uniform Random Variables

I would like to know if there is a general formula for calculating the sum of two uniform distributions $X$ and $Y$, where $X$ is uniformly distributed on $(-a,a)$ and $Y$ is uniformly distributed on ...
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1answer
231 views

Weakly bounded iff uniformly bounded in $E'$?

I have a problem: Suppose that $E$ be a normed space over $\mathbb{R}$ and $E= \{f: [0,1] \to \mathbb{R}\ \text{is continuous and such that}\ f|_{[0, \delta]}=0, \text{with}\ \delta=\delta(f)>0 ...
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1answer
61 views

Is $X_i$ in the following question uniform $\,k$-wise independent bits?

This is a homework question in the book named probability and computing. $13.9$ : suppose we are given m vectors $\overrightarrow v_1, \overrightarrow v_2 , ...
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2answers
259 views

Combining two identical uniform distributions

Say two random variables, $X$ and $Y$, are such that $X$ ~ $U(0,a)$ and $Y$ ~ $U(0,a)$. What will the pdf be for $Z$, where $Z=X-Y$?
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1answer
117 views

Generating points in rectangle

I am trying to generate $N$ points randomly and uniformly distributed in an $m \times n$ rectangle. How can this be done? I have tried to split the initial rectangle into as many rectangles i could, ...
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2answers
81 views

calculate probabilty, Uniform distribution

this is my first question so excuse my unknowing and mistakes: I was reading a book and just faced this thing: (1.4) $=P(X\gt Z/2)(Y-X)$ (1.5) $=P(2X\gt Z)(Y-X)$ (1.6) $=\min\{{2X,1\}}(Y-X)$ ...
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2answers
4k views

Probability density function of a product of uniform random variables

Let $z = xy$ be a product of two uniform random variables, with $x$ having the range $[a, b)$ and $y$ the range $[c, d)$. What is the probability density function of $z$, and how is it calculated?
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3answers
304 views

Find coordinates of n points uniformly distributed in a rectangle

I have a rectangle R of width W and height H. I have N points inside this rectangle. I need to find an algorithm to position my points in the rectangle in the most uniform way possible (no overlaps, ...
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1answer
163 views

A Problem on Uniform Probability Distribution

Consider three independent uniformly distributed (taking values between 0 and 1) random variables. What is the probability that the middle of the three values (between the lowest and the highest ...
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1answer
126 views

Why do prime numbers in modulo result in more uniform distributions?

Let us assume a sequence as follows: $S_{n} = (S_{n-1} * c_{1} + c_{2})\text{ mod } m$ This is the pseudorandom generator found in most programming languages' random function. It is known that a ...
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2answers
112 views

Showing the self energies of $N$ uniformly charged disks is proportional to $N^{3/2}$

How would I go about doing this? I assume it is some integral I have to solve, but I have no idea what. (Note:Not a physicist so please excuse incompetence with regard standard notation.) Context ...
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1answer
585 views

Distribution of Sum of Discrete Uniform Random Variables

I just had a quick question that I hope someone can answer. Does anyone know what the distribution of the sum of discrete uniform random variables is? Is it a normal distribution? Thanks!